3 2 2 1 1 3 3 1 1 2 2

(1)

(2)

2

86

11 8

1

^-1 ^{150 cm}

1

^-2 ⁸⁴ ⁷⁴

2

^-1 ^9.5

2

^-2

3

^-1

3

^-2

48000, 8000 8000 10

50 kg

1

^-1 ⁽ ⁾⁼

= =150(cm)

1

^-2 ⁽ ⁾⁼ ⁼ ^{=84( )}

x

=82, 336+x=410 x=74 74

2

^-1 ¹

2, 3, 4, 4, 5, 6, 7, 8 a= =4.5( ) 2

1, 3, 4, 5, 5, 6, 7, 10

b= =5( )

a+b=4.5+5=9.5

2

^-2 =8, 7+x=16 x=9

3

^-1 ⁵ ³ ⁵ ^a

2 3, 7 7+x

2

5+52 4+52 336+x

5

3364 90+82+86+78

4 750

5

110+147+153+161+179 5

D x

=80 338+x=400 x=62( )

=10 a+b+c+d+e=50

= = =28

x cm

=171.2, 2752-x=2568 x=184(cm)

50 kg

68, 69, 70, 72, 76, 78, 80, 82

=74(g)

n n

n { +1}

, 2

3 3

x

=83 x=86( )

, 92 3

86

A 9 7<a<12 6, 7, a, 12 =9 a=11 A, B

6, 7, 7, 8, 10, 11, 12, 13

=9

14 a=14

8, 9, 10, 12, 14, 14 ( )= 10+12=11

2 8+10

2 7+a

2 80+x

2 n 2

n 2

n+1 2 72+76

2 16_172-x

15

140 5 3(a+b+c+d+e)-10

5

(3a-2)+(3b-2)+ +(3e-2) 5

a+b+c+d+e 5

82+76+88+x+92 5

( )= =14=2( )

4+0+2+1+1+3+3 7 7

(3)

2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 8, 9 ( )=

= =6

( )= =6.5, ( )=8

( )>( )>( )

73 11 A

, 73 x 1

+1= , A+73+12=A+x x=85( )

12 aæ12

20 a…16

12…a…16

=2 a+b=12

, 2 a=2 b=2

a=2 , b=10, b=2 , a=10 , a>b a=10, b=2 a-b=10-2=8

80 50 (-30 ) 90 (+10 ) ,

(-6)+4+2+(-1)+3+a+b 7

A+x 12 A+7312

6+7 2 72 12

2+3+4+5+5+6+7+7+8+8+8+9 12

10 8

, , 4

15 , 16 , 18 , 18

x ,

=17.2 67+x=86 x=19( )

19 15+16+18_2+x

5

1

^-1

1

^-2 ⁷⁶

2

^-1 ² ^'2

2

^-2

3

^-1

3

^-2 ³⁸

4

^-1 ¹⁴⁰

2'3å5 kg

4

^-2 ^'1∂7.å6

5

^-1 ⁸⁴

5

^-2 ¹²⁰

6

^-1

6

^-2 ^{C, B}

82 -12, -2, 8, -2, 8 56

2'1å4 11 20.6

1

^-1 ⁰

(-4)+7+2+(-1)+x=0 x=-4

1

^-2 ^x

73 3

x-73=3 x=76( )

2

^-1 ⁽ ⁾⁼ ⁼ ^{=6( )}

( )= =2

( )='2( )

2

^-2 ⁼⁹ ^x=14

( )=

=60=12 5

(-3)¤ +(-1)¤ +5¤ +(-4)¤ +3¤

5 6+8+x+5+12

5

(-1)¤ +2¤ +(-2)¤ +0¤ +1¤

5 305 5+8+4+6+7

5

3 ( )=9.4 , ( )=7 , (

)=8 , 19

2+x+3+y+1=10 x+y=4

=21 3x+7y=16

, x=3, y=1

xy=3_1=3 ( )=

= =9.4( )

( )= =7( )

8 3 8

6+8 2 94 10

2+3_2+4+6+8_3+12+40 10

5_2+15_x+25_3+35_y+45_1 10

(4)

3

^-1 3, 5, a, 6, b 5

=5, a+b+14=25 a+b=11

, 5.4

=5.4 4+0+(a-5)¤ +1+(b-5)¤ =27

(a¤ -10a+25)+(b¤ -10b+25)+5=27 a¤ +b¤ -10(a+b)+28=0

a¤ +b¤ -10_11+28=0 a¤ +b¤ =82

3

^-2 ⁼⁶ ^x+y+z=18

=2 x¤ +y¤ +z¤ -12(x+y+z)+108=6 x¤ +y¤ +z¤ -12_18+108=6 x¤ +y¤ +z¤ =114

= =38

4

^-1 ⁽ ⁾⁼

= =75(kg)

( )

=

= =140

( )='1∂40=2'3å5(kg)

4

^-2 ⁽ ⁾⁼

= =10( )

( )

=

= ='1∂7.å6( )

5

^-1 ⁽ ⁾⁼

= =79( ) ( )=

= =84

5

^-2 ^{55 kg} ^x

2+4+x+4+2=20 x=8 84010

(-14)¤ _2+(-4)¤ _3+6¤ _4+16¤ _1 10

79010

65_2+75_3+85_4+95_1 10

352 20

(-8)¤ _2+(-4)¤ _4+0¤ _7+4¤ _6+8¤ _1 20

20020

2_2+6_4+10_7+14_6+18_1 20

420030

(-20)¤ _3+(-10)¤ _8+0¤ _9+10¤ _6+20¤ _4 30

2250 30

55_3+65_8+75_9+85_6+95_4 30

114 3 x¤ +y¤ +z¤

3

(x-6)¤ +(y-6)¤ +(z-6)¤

3 x+y+z

3

(3-5)¤ +(5-5)¤ +(a-5)¤ +(6-5)¤ +(b-5)¤

5 3+5+a+6+b

5

( )=

= =55(kg)

( )

=

= =120

6

^-1

6

^-2 ^C ^,

B C, B

2400 20

(-20)¤ _2+(-10)¤ _4+0¤ _8+10¤ _4+20¤ _2 20

1100 20

35_2+45_4+55_8+65_4+75_2 20

95 5

8

11 21

0

4+a+(-3)+b+(-1)+(-2)=0 a+b=2

3 x

(-3)+2+x+1+(-5)=0 x=5( )

90 5 3

95

(-3)+1+x+(-1)+0=0 x=3(kg)

( )= = =4

( )='4=2(kg)

( )= = =5( )

( )= = =2.5

( )= =8, 40+x=48

x=8 ( )=

= =5

=8 21+x+y=40 x+y=19

=2 9+1+1+x¤ -16x+64+y¤ -16y+64=10

x¤ +y¤ =16_19-129=175

(-3)¤ +(-1)¤ +1¤ +(x-8)¤ +(y-8)¤

5 5+7+9+x+y

5 306

(-4)¤ +(-1)¤ +2¤ +0¤ +0¤ +3¤

6 4+7+10+8+x+11

6

10 4 (-1)¤ +2¤ +(-2)¤ +1¤

4 20

4 4+7+3+6

4

205 (-3)¤ +1¤ +3¤ +(-1)¤ +0¤

5

æ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠

æ≠

(5)

6 x=8, y=6 81 90

=5'2

='7å0

x¡+x™+x£+ +x¡º=10 ( )=;1!0);=1

x¡¤ +x™¤ +x£¤ + +x¡º¤ =370 ( )

=

= = =36

'3å6=6

=78 450+75x+85y=1560 15x+17y=222 4+x+y+2=20 x+y=14

- _15 2y=12 y=6

y=6 x+6=14 x=8

( )=

= =81

5 x

80+75+85+90+95=85+70+95+85+x x=90

5 90

( )= 425 =85( ) 5

1620 20

(-13)¤ _4+(-3)¤ _8+7¤ _6+17¤ _2 20

65_4+75_x+85_y+95_2 20

360 10 370-2_10+10

10

x¡¤ +x™¤ +x£¤ + +x¡º¤ -2(x¡+x™+x£+ +x¡º)+10 10

(x¡-1)¤ +(x™-1)¤ +(x£-1)¤ + +(x¡º-1)¤

10 (x+y)¤ =x¤ +y¤ +2xy 19¤ =175+2xy

186=2xy xy=93

( )= = =5

( )=

= =1

( )='1=1

(-2)¤ _1+(-1)¤ _2+0¤ _3+1¤ _4=10

2 1 0

( )= = =8

2+4+a+1+1=10 a=2 ( )=

= =7( )

( )

=

= =5.8

=

=' ∂ ∂ ( )=

= =72( )

( )

=

='1å2å1=11( )

=5 a+b+c+d+e=25

=3¤

( )=10_{ }+ º;;=50+2=52

( )

=

=10

=10_3=30

(a-5)¤ +(b-5)¤ +(c-5)¤ +(d-5)¤ +(e-5)¤

5

(10a-50)¤ +(10b-50)¤ +(10c-50)¤ +(10d-50)¤ +(10e-50)¤

5 a+b+c+d+e

5

(a-5)¤ +(b-5)¤ +(c-5)¤ +(d-5)¤ +(e-5)¤

5 a+b+c+d+e

5

(-17)¤ _3+(-7)¤ _6+3¤ _6+13¤ _4+23¤ _1 20

1440 20

55_3+65_6+75_6+85_4+95_1 20

{( )¤ _ }

( )

5810

(-3)¤ _2+(-1)¤ _4+1¤ _2+3¤ _1+5¤ _1 10

70 10

4_2+6_4+8_2+10_1+12_1 10

80 10 6_10+4_5

10

'63 2'6

3 1010

(-2)¤ _1+(-1)¤ _2+0¤ _3+1¤ _4 10

50 10 3+4_2+5_3+6_4

10

( )=

= =81( )

( )

=

= =84

( )='8å4=2'2å1( )

A B '7å0

'1å0 5 5 840

10

(-16)¤ _1+(-6)¤ _4+4¤ _3+14¤ _2 10

810 10

65_1+75_4+85_3+95_2 10

æ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠

æ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ —

æ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ –

(6)

( )

=

= ='5å0=5'2( )

( )

= = 350 ='7å0( )

5 0¤ +(-15)¤ +10¤ +0¤ +5¤

5 250

5

(-5)¤ +(-10)¤ +0¤ +5¤ +10¤

æ≠ ≠ ≠ ≠ 5 ≠ ≠ ≠

æ≠

æ≠ ≠ ≠ ≠ ≠ ≠

1

^-1

1

^-2

2

^-1

2

^-2 ⁴^'1å0

3

^-1 ^{5 cm}

3 cm 28 cm

3

^-2 ¹⁶⁹

4

^-1 ^{49 cm¤}

4

^-2

5

^-1

5

^-2 ^{5 cm}

6

^-1

6

^-2 ³

5 4'5

1

^-1 ^ACD ^x="1√3√¤ -≈5¤ ='1∂44=12

ABD y="2√0√¤ -≈x¤ ="2√0√¤ √-√≈1≈2¤ ='2∂56=16 x+y=12+16=28

1

^-2 ^ABC ^AB”="1√0√¤ -≈6¤ ='6å4=8 ABD x="√AB”¤ √+BçD”¤ ="8√¤ √+√1≈2¤

='2ß0å8=4'1å3

2

^-1 ^AC”="1√¤ +1¤ ='2, AD”="(√'2)¤ ç+≈1Ω¤ ='3 AE”="(√'3)¤ ç+≈1Ω¤ =2, AF”="2¤√ +1¤ ='5

AG”="(√'5)¤¤ ç+≈1Ω¤ ='6

2

^-2 ^A ^BC” ^H

ABH AH”="8√¤ -2¤ =2'1å5 CD”=AH”=2'1å5

BCD BD”="1√0¤ +√(2'1åç5)Ω¤ =4'1å0

3

^-1 AEH™ BFE™ CGF™ DHG(SAS )

HEF=90 EFGH

EH”¤ =25 EH”=5(cm) EFGH

EH”=EF”=GF”=GH”=5(cm)

AEH AH”="√5¤√ -4¤ ='9=3(cm)

AEH™ BFE™ CGF™ DHG AH”=BE”=CF”=DG”=3(cm)

AB”=BC”=CD”=DA”=4+3=7(cm)

ABCD 4_7=28(cm)

3

^-2 ^AEH ^EH”="5√¤ +√1≈2¤ ='1∂69=13 EFGH=13¤ =169

4

^-1 ^BCG ^BG”="1√7¤ -≈8≈¤ ='2∂25=15(cm) FG”=BG”-BF”=15-8=7(cm)

EFGH=7¤ =49(cm¤ )

4

^-2 ^PQRS ^PS”=^'4å9=7(cm)

ASD

AS”=AP”+PS”=DS”+PS”=5+7=12(cm) AD”="1√2¤ +≈5Ω¤ =13(cm)

ABCD

ABCD=AD”¤ =169(cm¤ )

5

^-1 EBC™ ABF(SAS ) EBC= ABF

, EBC EBA, ABF BFL

EBA= EBC= ABF= BFL

5

^-2 ADEB= BFGC+ ACHI 34=BC”¤ +9, BC”¤ =25

BC”=5(cm)( BC”>0)

6

^-1 ^{2¤ =1¤ +(}^'3)¤ ⁽'1å7)¤ =1¤ +4¤

(2'2)¤ =2¤ +2¤ 5¤ =3¤ +4¤

6¤+4¤ +('1å5)¤

6

^-2 (x+2)¤ =(x+1)¤ +3¤

x¤ +4x+4=x¤ +2x+10, 2x=6 x=3

2

20 72 cm¤

6 cm¤

10

AB”=BC”=9 cm, CF”=3 cm

ABF AF”="√AB”¤ √+BçF”¤ ="9¤√ +1ç2¤ =15(cm) ABC BC”="1√3¤√ -≈5¤ ='1∂44=12

CD”=;2!; BC”=;2!;_12=6 ADC x="5√¤ +6¤ ='6å1

(7)

AB”=x

BP”='2x, “CP='3x, “DP=2x, “EP='5x '5x=2'5 x=2

ABC AC”=AD”="1√¤ +1¤ ='2 ADE AE”=AF”="1√¤ +(√'2)¤ ='3 AFG AG”=AH”="1√¤ +(√'3)¤ =2

D BC” H

DH”=AB”=4(cm)

BH”=AD”=2(cm) CH”=5-2=3(cm) CDH CD”="3√¤ +4¤ =5(cm)

ABCD AD”='3å6=6

AH”=6-2=4

AEH EH”="4√¤ +2¤ =2'5 EFGH=EH”¤ =20

ABQ=;2!;_'3_1=

“AQ='3 PQRS '3-1

.

PQRS=('3-1)¤ =4-2'3 CBH= ABH= CBG= LBG

BHIC= LMGB

BFMN= ADEB BFMN=16(cm¤ )

BFN=;2!; BFMN BFN=;2!;_16=8(cm¤ ) ABE™ CDB BE”=DB”

EBD=90 EBD

EBD=;2!;_EB”¤ =40 EB”=4'5(cm)( EB”>0)

EA”="(√4'5√)√¤ -4¤ =8(cm)

ACDE=;2!;_(8+4)_12=72(cm¤ ) 4¤ =3¤ +('7)¤ ( )

4¤ =(2'3)¤ +2¤ ( ) (2'3)¤ =2¤ +(2'2)¤ ( ) i) x , x¤ =5¤ +12¤ =169

x=13( x>0)

ii) 12 , 12¤ =5¤ +x¤ , x¤ =119 x='1∂19( x>0)

ABE AE”="1√5¤ -ç12¤Ω =9(cm) ED”=12-9=3(cm)

ABEª DFE AA

9 3=12 FD” FD”=4(cm) '32

16 6 6'5 88 cm¤ 5 cm

13 cm ;1^3); cm

ABC BC”="2√0¤ -ç12Ω¤ =16 CD”=x BD”=16-x

20 12=(16-x) x, 20x=12(16-x) 20x=192-12x, 32x=192 x=6

ADC AD”="6¤√ +1≈2Ω¤ =6'5

A, D BC

E, F

AD”=EF”=5(cm)

BE”=;2!;_(17-5)=6(cm)

ABE AE”="1√0¤ -≈6Ω¤ ='6å4=8(cm) ABCD=;2!;_(5+17)_8=88(cm¤ )

ABGF AE”

AE”¤ =25 AE”=5(cm)( AE”>0) AED AD”="1√2¤ +≈5¤Ω ='1∂69=13(cm) EF”=x AED=;2!;_12_5=;2!;_13_x 13x=60 x=;1^3);(cm)

FED=;2!;_3_4=6(cm¤ ) AB”=18-12=6(cm)

ABC

AC”="1√0¤ -Ω6Ω¤ =8(cm) x=20-8=12(cm)

ABC BC”="√4¤ +≈6Ω¤ ='5å2=2'1å3(cm) BDEC=(2'1å3)¤ =52(cm¤ )

FDE=;2!; BDEC=;2!;_52=26(cm¤ ) (x+4)¤ =x¤ +(x+2)¤ , x¤ +8x+16=x¤ +x¤ +4x+4 x¤ -4x-12=0, (x+2)(x-6)=0

x=6( x>0)

10 .

20 cm

12 cm 18 cm

10 cm x cm

A

B 6 cm C

B

A D

F C E

5 cm

10 cm

6 cm 5 cm

(8)

1

^-1

1

^-2 ²'1å3<a<10

2

^-1

2

^-2

3

^-1 ⁷

3

^-2 ^{;1^3); cm}

4

^-1

4

^-2

5

^-1

5

^-2

6

^-1 ^13p

6

^-2

7

^-1

7

^-2 ^{º;; cm}

x= ™5¢;;, y= ™;; 3'2 cm ™2∞;;p

1

^-1

8-5<x<8+5 3<x<13 x>8 8<x<13

x¤ <5¤ +8¤ , x¤ <89 0<x<'8å9

, 8<x<'8å9

1

^-2

6-4<a<6+4 2<a<10 a>6 6<a<10

a¤ >4¤ +6¤

a>2'1å3

, 2'1å3<a<10

2

^-1 11¤ <7¤ +9¤ ( ) 3¤ =('5)¤ +2¤ ( ) 4¤ >2¤ +3¤ ( ) 6¤ >('1å4)¤ +4¤ ( ) 13¤ =5¤ +12¤ ( )

2

^-2 ⁽'1å0)¤ >2¤ +('5)¤

3

^-1 ^BC”="3√¤ +4¤ ='2å5=5

3¤ =x_5 x=;5(;, 4¤ =y_5 y=

y¤ -x¤ ={ }¤ -{;5(;}¤ =7

3

^-2 ^ABC

AC”="1√3¤ -ç5¤ ='1∂44=12(cm) , AB”_AC”=BC”_AH”

5_12=13_AH”

AH”=;1^3);(cm)

4

^-1 ^{AB”¤ +}“CD¤ =“AD¤ +“BC¤

AB”¤ +“CD¤ =4¤ +8¤ =16+64=80

4

^-2 ^{AB¤” +}“CD¤ =“AD¤ +“BC¤

('1å3)¤ +x¤ =(2'2)¤ +5¤ , x¤ =20 x=2'5( x>0)

5

^-1 ^{AP”¤ +}“CP¤ =“BP¤ +“DP¤ 5¤ +3¤ =4¤ +“DP¤

“DP¤ =18 “DP=3'2(cm)( “DP>0)

5

^-2 PA”¤ +PC”¤ =PB”¤ +PD”¤

PB”¤ -PA”¤ =PC”¤ -PD”¤ =4¤ -('7)¤

=16-7=9

6

^-1 ^(R ⁾⁼^;2!;_p_{ ^}^{¤ = p}

P+Q=R

P+Q+R=2R=2_ p=13p

6

^-2⁽ ^{)= ABC}

=;2!;_6_8

=24(cm¤ )

7

^-1 ^{DB”=x cm} DC”=DA”=(9-x) cm DBC (9-x)¤ =x¤ +6¤

81-18x+x¤ =x¤ +36, 18x=45 x=;2%;(cm)

7

^-2 ^ABP ^{AP”=10 cm}

BP”="1√0¤√ -6¤ ='6å4=8(cm) PC”=10-8=2(cm) PQ”=DQ”=x QC”=6-x

PCQ x¤ =2¤ +(6-x)¤

x¤ =4+36-12x+x¤ , 12x=40 x=PQ”= º;;(cm)

2'1å3 2

4'3 cm

14p ;8%;

4'5 5

(x+3)¤ >x¤ +9¤ x>12 ('5)¤ =1¤ +2¤ ( ) (3'6)¤ >3¤ +6¤ ( ) 10¤ >4¤ +7¤ ( )

(9)

9¤ <7¤ +8¤ ( )

10¤ =(2'5)¤ +(4'5)¤ ( )

C<90 c¤ <a¤ +b¤ . c¤ =a¤ +b¤ C=90 .

C A B

. AC”¤ =CH”_BC” 4¤ =2_BC”

BC”=8(cm) , BH”=8-2=6(cm)

AB”¤ =BH”_BC” AB”¤ =6_8=48 AB”=4'3(cm)( AB”>0) AB”¤ =BH”_BC” 6¤ =BH”_12

BH”=3(cm)

H”M”=B’M”-BH”=;2!;_12-3=3(cm) AH”¤ =BH”_CH”” AH”¤ =3_9=27

AH”=3'3(cm)( AH”>0) AHM=;2!;_H”M”_AH”

=;2!;_3_3'3= (cm¤ )

4x-2y+8=0 y 4 ,

x -2

AB”="2√¤ +4¤ ='2å0=2'5 2_4=OH”_2'5

OH”=

BC”¤ +DE”¤ =BE”¤ +CD”¤ , 8¤ +DE”¤ =7¤ +6¤

DE”¤ =21 DE”='2å1( DE”>0) DOC CD”="7¤√ +≈2¤ ='5å3 AB”¤ +CD”¤ =AD”¤ +BC”¤

5¤ +('5å3)¤ =AD”¤ +8¤ , AD”¤ =14

“AD='1å4( AD”>0) 8¤ +(x+2)¤ =(2'2å1)¤ +x¤

64+x¤ +4x+4=84+x¤ , 4x=16 x=4 AB”, BC”, AC”

S¡, S™, S£

S¡=;2!;_p_(2'3)¤ =6p, S™=8p S£=S¡+S™ S£=6p+8p=14p

ABC AB”="(√3'5)¤√ -3¤ ='3å6=6(cm)

( )= ABC

=;2!;_3_6=9(cm¤ ) 4'5

5

9'3 2

2 ;2%; ;5*; 4"5 cm 3 cm 6 cm¤

ABC AG”¤ =BG”_CG”

AG”¤ =1_4 AG”=2( AG”>0)

M ABC

A’M”=B’M”=C’M”=;2!; BC”=;2%;

AMG AG”¤ =AH”_A’M”

2¤ =AH”_;2%; AH”=;5*;

ABE™ C'DE BE”=DE”=x cm AE”=C’'E”=(6-x) cm

ABE 3¤ +(6-x)¤ =x¤ x= ∞;;(cm)

(5, 7, 8), (5, 7, 11), (5, 8, 11), (5, 11, 13), (7, 8, 11), (7, 8, 13), (7, 11, 13), (8, 11, 13) 8 (5, 7, 11), (5, 8, 11), (5, 11, 13), (7, 8, 11), (7, 8, 13) 5

;8%;

DE”

ADE DE”="3√¤ +4¤ =5 ABE BE”="1√0¤ +ç4¤ =2'2å9

DE”¤ +BC”¤ =BE”¤ +CD”¤

5¤ +BC”¤ =(2'2å9)¤ +CD”¤

BC”¤ -CD”¤ =116-25=91

S¡+S™= ABD S£+S¢= BCD

S¡+S™+S£+S¢

= ABD+ BCD

= ABCD=8(cm¤ )

ABD' BD”'’="1√5¤ -ç9¤ =12(cm) CD”'’=15-12=3(cm)

CE”=x E”=DE”=9-x

D'CE 3¤ +x¤ =(9-x)¤ x=4(cm) ABD'=;2!;_9_12=54(cm¤ ) D'CE=;2!;_3_4=6(cm¤ )

ABD' D'CE=54 6=9 1

2 4

B O

H A

A D

B C

4 2

(10)

1

^-1

1

^-2 ⁶^'5

2

^-1 ⁵^{'2 cm}

2

^-2

3

^-1

3

^-2 ;5&; cm

4

^-1

4

^-2 ^{4 3}

5

^-1

5

^-2

6

^-1 ^{60 cm¤}

6

^-2 ²^{'6 cm}

4'5 cm 3'2 cm 2 cm 4 cm 4'3 cm h=3, S=12

1

^-1 ⁽ ⁾⁼"1≈3≈¤ √-√12¤ ='2å5=5(cm)

( )=12_5=60(cm¤ )

1

^-2 2k, k(k>0)

"(√2k)¤ ç+≈kΩ¤ =15, 5k¤ =225 k¤ =45 k=3'5( k>0)

2_3'5=6'5

2

^-1 ^x

'2x=10 x=5'2(cm)

2

^-2 ^a

'2a=8'6 a=8'3(cm)

8'3_4=32'3(cm)

3

^-1 ^BD”="6√¤ +8¤ ='1∂00=10(cm)

“AB_“AD=“BD_A’H” 6_8=10_“AH 10“AH=48 “AH=4.8(cm)

3

^-2 ^BD”="3¤√ +4¤ =5(cm) AB”¤ =BE”_BD”

3¤ =BE”_5 BE”=;5(;(cm) DF”=;5(;(cm) EF”=5-2_;5(;=;5&;(cm)

4

^-1 ^ABC ^a

AH”= a=4'3 a=8(cm) ABC

_8¤ =16'3(cm¤ )

4

^-2 ^AD”=øπAB”¤∑ ∑-∑“B∑D¤ =Æa¤ …-{;…2!;a}¤ =Æ;4#;¬a¤ = a ABC= a¤ , ADE= _{ a}¤ = a¤

ABC ADE=4 3

5

^-1 ^AC” ^B=60 ^AB”=BC”

ABC ACD

ABCD a

ABCD=2 ABC=2_ a¤ =50'3 a=10(cm)( a>0)

5

^-2 ⁶

6_{ _6¤}=54'3(cm¤ )

6

^-1 ^A ^BC” ^H

BH”=CH”=;2!;BC”=;2!;_10=5(cm)

ABH AH”="1√3¤ -≈5Ω¤ =12(cm) ABC=;2!;_10_12=60(cm¤ )

6

^-2 ^BH”=x ^CH”=6-x

5¤ -x¤ =7¤ -(6-x)¤ , 25-x¤ =49-36+12x-x¤

12x=12 x=1(cm)

AH”="5√¤ -1¤ ='2å4=2'6(cm) '34

'34

3'3 '3 16

'3 2 '3 4

4

'32 '34

'32

6'5 cm

;; cm cm¤

10('5+1)cm cm¤

9'3 4

( )="1√7¤ -ç1ç5¤ ='6å4=8(cm)

( )=2_(15+8)=46(cm)

S¡=2p cm¤ , S¡ S™=1 4 S™=8p cm¤

S£=S¡+S™ S£=2p+8p=10p(cm¤ ) BC”=2x

;2!;_p_x¤ =10p, x¤ =20 x=2'5(cm)( x>0) BC”=2_2'5=4'5(cm)

BE”=x cm DE”=AE”=(8-x) cm BD”=DC”=;2!;_8=4(cm)

BDE (8-x)¤ =x¤ +4¤

64-16x+x¤ =x¤ +16, 16x=48 x=3(cm) DBE=;2!;_BD”_BE”=;2!;_4_3=6(cm¤ )

(11)

r (2r)¤ +(2r)¤ =20¤ , 8r¤ =400 r¤ =50 r=5'2( r>0)

( )=pr¤ =p_(5'2)¤ =50p a

26¤ =(5a)¤ +a¤ a='2å6(cm)( a>0)

“AC="(√2√'2å6)√¤ √+('√2å6)¤ ='1å3å0(cm)

ABCD x '2x=6'2

x=6(cm)

ECFG y '2y=12'2

y=12(cm)

DCF

DF”="x√¤ +y¤ ='3∂6ƒ+14å4=6'5(cm) BD”="9√¤ +1≈2Ω¤ =15(cm)

AB”_AD”=BD”_AH” 9_12=15_x x= (cm)

AD”¤ =DH”_DB” 12¤ =y_15 y= (cm)

x+y= + =;; (cm) a

_a¤ =64'3, a¤ =256 a=16(cm)( a>0) ( )= _16=8'3(cm)

a h, S

h= a, S= a¤

AD”= _4=2'3(cm)

AG” 1 AG”= _2'3= (cm)

( )

= ABC+ DEF- GEC=2 ABC- GEC

=2_{ _4¤}-{ _2¤}=8'3-'3=7'3(cm¤ )

ABC 4 cm

AD”= _4=2'3(cm)

ADE 2'3 cm

AF”= _2'3=3(cm)

AFG 3 cm

AFG= _3¤ = 9'3(cm¤ ) '3 4

4 '32

'32

'34 '34

4'3 2 3

3 '32

'34 '32

'32 '34

6 .

a 6_ a¤=96'3, a¤ =64

a=8(cm)( a>0)

6_8=48(cm) BH”=CH”=;2!;_10=5(cm)

ABC AB”="5√¤ +1≈0Ω¤ =5'5(cm) ABC

5'5+5'5+10=10('5+1) cm

A BC” H

BH”=x CH”=14-x

13¤ -x¤ =15¤ -(14-x)¤ , 28x=140 x=5(cm)

AH”="√13√¤ -5¤ ='1∂44=12(cm) ABC=;2!;_14_12=84(cm¤ ) BD”="1√2¤ +ç16¤Ω =20(cm)

AB”_AD”=BD”_AE” 12_16=20_AE”

AE”= (cm)

AB”¤ =BE”_BD”” 12¤ =BE”_20 BE”= (cm)

EF”=BD”-2BE”=20-2_ = ™5•;;(cm) AECF=2 AEF

=2_{;2!;_ ™5•;;_ }

= (cm¤ )

ABC= APB+ BPC+ CPA

_2¤ =;2!;_2_“PD+;2!;_2_“PE+;2!;_2_“PF

“PD+“PE+“PF='3 r cm

r 12 cm

r= _12=6'3

p_(6'3)¤ =108p(cm¤ )

A EF” H

A EF” AH”

AE”=AF”="8√¤ +2¤ ='6å8=2'1å7

EF”="6¤√ +6¤ ='7å2=6'2 EH”=3'2 AEH

AH”="(√2'1å7√)¤ -√(√3√'2)¤ ='5å0=5'2 '32

'34 '34

r cm 12 cm

(12)

1

^-1

1

^-2 ⁴⁽^'3+1)

2

^-1 ²^{'3 cm}

2

^-2

3

^-1

3

^-2 ^{60 cm¤}

4

^-1

4

^-2 ^-2

5

^-1

5

^-2

29

6

^-1

6

^-2 ⁵

7

^-1

7

^-2 ²^'4å1

x=4 cm, y=4'2 cm x=4'3 cm, y=4 cm x=6'3, y=3'6 '5 '2å6 2'5

1

^-1 ^ABC ^x=4^'3

BCD 4'3 y='3 2 y=8 xy=4'3_8=32'3

1

^-2 ^A ^BC” ^H

ABH 8 BH”=2 '3 BH”=4'3 8 AH”=2 1 AH”=4

ACH 4 HC”=1 1 HC”=4

BC”=BH”+HC”=4'3+4=4('3+1)

2

^-1 ^ABC ^{8 BC”=2} ^'3 ^BC”=4^'3(cm)

BCD 4'3 BD”=2 1 BD”=2'3(cm)

2

^-2 ^ABH ^{4 BH”=2 1} ^BH”=2(cm)

4 AH”=2 '3 AH”=2'3(cm) ACH CH”=8-2=6(cm) AC”="(√2'3)√¤ +6¤ =4'3(cm)

3

^-1 ^EBF ^EF”='1∂44=12(cm) EB” 12='3 2 EB”=6'3(cm) BF” 12=1 2 BF”=6(cm)

AB”=AE”+EB”=BF”+EB”

=6+6'3=6(1+'3)(cm)

3

^-2 ^A ^BC” ^H

ABH 4'3 AH”=2 '3 AH”=6(cm) ABCD=10_6=60(cm¤ )

4

^-1 "3√¤ +4¤ =5 "4√¤ +0¤ =4

"1√¤ +2¤ ='5 "0¤√ +4¤ =4

"2√¤ +1¤ ='5

4

^-2 ^AB”="(√3-a√)¤ +√(√-√4√-1)Ω¤ =5'2 a¤ -6a+9+25=50, a¤ -6a-16=0 (a-8)(a+2)=0 a=8 a=-2

A 2 a=-2

5

^-1 ^OA”="1√¤ +2¤ ='5, OB”="(√-2)√¤ +3¤ ='1å3 AB”="(√-2√-1)√¤ +(√3-2≈)Ω¤ ='1å0

OB” , OB”¤ <OA”¤ +AB”¤

OAB .

5

^-2 ^AB”="(√-4√-6)√¤ +(√0+4≈)¤Ω ='1∂16=2'2å9 BC”="(√3+4√)¤ +√(3-ç0)Ω¤ ='5å8

CA”="(√3-6√)¤ +√(3+≈4≈)Ω¤ ='5å8 AB”¤ =BC”¤ +CA”¤ , BC”=CA”

4'2cm 4 cm (48p-64) cm¤

8'2 cm 2'6 cm 6'3 cm¤ 3'7 cm '7å9 cm

8cm 8'2cm

;2!;_8'2=4'2(cm) 8cm 8cm

;2!;_8=4(cm)

= - +

=p_(4'2)¤ -8¤ +p_4¤

=48p-64(cm¤ ) ABC

BC”="(√4'6)√¤ +(√4'2)Ω¤ ='1∂28=8'2(cm) AB”_AC”=BC”_AD”

4'6_4'2=8'2_AD” AD”=2'6(cm) ADE= _(2'6)¤ =6'3(cm¤ )

CH”=x BH”=10-x

AH”¤ =12¤ -(10-x)¤ =8¤ -x¤ x=1(cm) ACH AH”="8√¤ -1¤ ='6å3=3'7(cm) B’M”=CM”=;2!;BC”=;2!;_10=5(cm) M”H”=M”C”-CH”=5-1=4(cm)

AMH A’M”="4¤√ +(√3'7)Ω¤ ='7å9(cm) '34

-2-1 1 2 3 B

A x y

2

-2 -1 -3 1 O

(13)

ABC C=90 . ABC=;2!;_'5å8_'5å8=29

6

^-1y=(x¤ -4x+4-4)+2=(x-2)¤ -2 P(2, -2)

x=0 , y=2 Q(0, 2) PQ”="(√2-0√)¤ +√(-2√-2)Ω¤ =2'5

6

^-2y=2x¤ +4x+1=2(x¤ +2x+1-1)+1

=2(x+1)¤ -1

(-1, -1)

(-1, -1) A(2, 3)

"(√2+1√)¤ +√(3+√1)¤ ='2å5=5

7

^-1 ^B ^x

B' B'(5, -1)

(AP”+BP” )

=A’B'”

="(√5-1√)¤ +√(-1√-ç3)Ω¤

='3å2=4'2

7

^-2 ^A ^BC”

A', A' BC”

DC” D'

DA'D' A’'D”="1√0¤ +Ω8Ω¤ ='1∂64=2'4å1

4 cm 6+10'3

6'2 2'1å3 (8+4'3) cm B(4, 8)

ABC AB” BC”=1 '3 2 BC”=1 '3 BC”=2'3(cm)

BDC BC” CD”='2 1 2'3 CD”='2 1 CD”='6(cm)

45 , 45 , 90 1 1 '2 , 30 , 60 , 90

1 '3 2 .

ACD AD” AC”=1 '2

2'6 AC”=1 '2 AC”=4'3(cm) ABC AC” BC”='3 2

4'3 BC”='3 2 BC”=8(cm) HBC BH” BC”='3 2 6 BC”='3 2 BC”=4'3(cm)

ABC BC” AC”='3 1 4'3 AC”='3 1 AC”=4(cm)

A=60 BAD= DAC=30

ABC AB “AC=2 1, 2'3 AC”=2 1

“AC='3(cm)

ADC AD “AC=2 '3, AD '3=2 '3 AD”=2(cm)

x

ABC AC” BC”=1 '2 AC” x=1 '2

AC”= x

20 cm x+x+ x=20, ('2+1)x=20

x= = =20('2-1)(cm)

A,

D BC

E, F ABE

AB” BE”='2 1

2'6 BE”='2 1 BE”=2'3 DFC DF” FC”='3 1 2'3 FC”='3 1 FC”=2

ABCD

;2!;_{4+(2'3+4+2)}_2'3

=(10+2'3)_'3=6+10'3

'5 '1å3 '3å7 3'2 '3å4

PQ”¤ =(a-2)¤ +(-1-2)¤ =45 a¤ -4a+4+9=45, a¤ -4a-32=0 (a-8)(a+4)=0 a=-4( a<0)

x C(x, 0)

AC”=BC”

"(√x+1√)¤ +√(0-ç3)Ω¤ ="(√x-2√)¤ +√(0-ç4)Ω¤

x¤ +2x+10=x¤ -4x+20, 6x=10 x=;3%;

20('2-1) ('2+1)('2-1) 20

'2+1 '22 '22

'22

y

x 4

2

O 2 P 4

B' B A

6 2

2 10 2

A B

D

P C

A' D'

45

A C

B 20 cm

2'6 2'3

45 60

4

4 2 C

B

D

E F A

(14)

AB”="(√2+1√)¤ +√(1-ç3)¤Ω ='å1å3 BC”="(√4-2√)¤ +√(4-ç1)¤Ω ='1å3 CA”="(√4+1√)¤ +√(4-≈3Ω)Ω¤ ='2å6 CA”¤ =AB”¤ +BC”¤ , AB”=BC”

B=90 .

ABC=;2!;_'1å3_'1å3=

y=x+4 y=;2!;x¤

x+4=;2!;x¤ , x¤ -2x-8=0, (x-4)(x+2)=0

x=-2 x=4

, A(-2, 2), B(4, 8)

“AB="(√4+2√)¤ +√(8-√ç2)¤ ='7å2=6'2 A y

A'(-2, 5) AP”=A’'P”

AP”+BP”

A’'B”="{√4-(√-2)√}¤ +(√1-5≈)≈¤

='5å2=2'1å3

ADC AD” 4=2 1 AD”=8(cm) DC” 4='3 1 DC”=4'3(cm)

ADC= ABD+ BAD

30 =15 + BAD BAD=15 , ABD BD”=AD”

BD”=AD”=8(cm)

BC”=BD”+DC”=8+4'3(cm)

A CE” H

ACE=60 ACH “AC “AH=2 '3

4 AH”=2 '3 AH”=2'3(cm) ACE=;2!;_6_2'3=6'3(cm¤ )

B {a, ;2!;a¤ } OA”¤ +AB”¤ =OB”¤

(-2)¤ +2¤ +{ a-(-2) }¤ +{;2!;a¤ -2}¤ =a¤ +{;2!;a¤ }¤

2a¤ -4a-16=0, a¤ -2a-8=0, (a+2)(a-4)=0

a=-2 a=4

B 1 a=4

B(4, 8)

P Q BC”, AD”

P', Q'

P’'Q'”

P’'Q'”="1√0¤ √+√10¤ ='2∂00=10'2(cm)

72('3-1) 4 cm 4'3 cm 8'3 cm¤

AB”=5'2, BC”=2'1å0, CA”='1å0 C=90 10

ADC AC” DC”=1 1 AC” 12=1 1 AC”=12

ABC BC” AC”='3 1 BC” 12='3 1 BC”=12'3

BD”=BC”-DC”=12'3-12

ABD=;2!;_(12'3-12) 12=72('3-1) ABC AB” 8=1 2 AB”=4(cm) ABC AC” 8='3 2 AC”=4'3(cm)

ABC

( )= ABC=;2!;_4_4'3

=8'3(cm¤ ) AB”="(√-3√-2)¤√ +(√0-5≈)≈¤ ='5å0=5'2

BC”="(√3+3√)¤ +√(2-ç0)≈Ω¤ ='4å0=2'1å0 CA”="(√3-2√)¤ +√(2-ç5)≈¤ ='1å0

“AB¤ =“BC¤ +“CA¤

ABC C=90 .

ABC=;2!;_2'1å0_'1å0=10

'7å0 cm 4'3 cm 2'7 cm

cm‹ 4 cm 12p cm‹

'1∂37 cm 32'7

3

1

^-1

1

^-2 ^cm¤

2

^-1 ^{27 cm‹}

2

^-2

3

^-1

3

^-2 ^{12 cm}

4

^-1 ⁸^{'3å4 cm¤}

4

^-2 ⁶⁴^{'2 cm‹}

5

^-1 ³^{'3 cm} ⁹^{'3p cm‹}

5

^-2

6

^-1

6

^-2

5'1å1 2

x A'

y

1 2 O

-2 4

A

P B 5

2 cm

2 cm Q' A

P'

C D 6 cm

10 cm B

(15)

1

^-1 ^h

"4√¤ +3√¤ +h¤ =5'2, 25+h¤ =50, h¤ =25 h=5( h>0)

1

^-2^AE”=x

"4√¤ +3√¤ +x¤ =6, 25+x¤ =36, x¤ =11 x='1å1(cm)( x>0)

, EG”="3√¤ +4¤ =5(cm)

AEG=;2!;_5_'1å1= (cm¤ )

2

^-1 ^a

'3a=3'3 a=3(cm) ( )=3‹ =27(cm‹ )

2

^-2 ^DAB “BD="9¤√ +9¤ ='1∂62=9'2(cm) DAE “DE="9¤√ +9¤ ='1∂62=9'2(cm) BEF “BE="9¤√ +9¤ ='1∂62=9'2(cm)

( DEB )=“BD+“DE+“BE

=9'2+9'2+9'2

=27'2(cm)

3

^-1 ^a

a=2'3 a=3'2(cm)

( )= _(3'2)‹ =9(cm‹ )

3

^-2 ^a

a‹ =144'2 a=12(cm)

4

^-1^BD”="8√¤ +8¤ =8'2(cm)

HD”=;2!;BD”=;2!;_8'2=4'2(cm) OH”="1√0¤ -√(4'√2)¤ ='6å8=2'1å7(cm)

OBD=;2!;_BD”_OH”

=;2!;_8'2_2'1å7=8'3å4(cm¤ )

4

^-2 ^OCH ^CH”="(√5'2)√¤ -(√3'2)Ω¤ =4'2(cm) AC”=2CH”=2_4'2=8'2(cm)

ABCD

a '2a=8'2 a=8(cm)

( )=;3!;_8¤ _3'2=64'2(cm‹ )

5

^-1 ^r

2p_r=6p r=3(cm) AOB

( )="6√¤ -3¤ ='2å7=3'3(cm)

( )=;3!;_(p_3¤ )_3'3=9'3p(cm‹ ) '212

'212 '63

5'1å1 2

5

^-2 ^AC”="8√¤ -4¤ ='4å8=4'3(cm) ( )=;3!;_(p_4¤ )_4'3

= p(cm‹ )

6

^-1 ⁽ ⁾⁼“AG="6¤√ +9¤

='1å1å7

=3'1å3(cm)

6

^-2 ^OABC

AC” OB”

A’M”=C’M”, O’M”=B’M”

A’M”= _2'3=3(cm)

( )=AC”=2A’M”=2_3=6(cm) '32

64'3 3

5'3 cm 9'2 cm¤

5p cm¤ 10p

96 cm¤

EFG

EG”="8√¤ +6¤ =10, EO”=;2!;EG”=;2!;_10=5 AEO AO”="8√¤ +5¤ ='8å9

AG”="8√¤ +6√¤ +1ç0¤ ='2∂00=10'2(cm) EG”="8√¤ +6¤ ='1∂00=10(cm)

AEG AE”_EG”=AG”_E’I’

10_10=10'2_E’I’ E’I’=5'2(cm)

"1√0¤ +√10¤ √+√10¤ =10'3(cm)

;2!;_10'3=5'3(cm) .

AMGN A’M”=MÚG”=GN”=NÚA”=2'5(cm)

“AG=4'3(cm), M”N”=“FH=4'2(cm) AMGN=;2!;_“AG_M”N”

=;2!;_4'3_4'2=8'6(cm¤ )

4 cm A

B 8 cm

l

C G

F A

6 cm

5 cm B 4 cm D

2'3 cm

A C

B M

O

(16)

;3$; 2'3 6 cm 216p cm‹ 90 8'5 cm

2'3 3 A’M”=D’M”= _18=9'3(cm)

M”H”=;3!;_9'3=3'3(cm) AMH

AH”="(√9'3)¤√ -(√3√'3)¤ ='2∂16=6'6(cm) AMH=;2!;_3'3_6'6=27'2(cm¤ ) B’M”=C’M”= _6

=3'3(cm) BH”=CH”=3(cm)

M”H”="(√3'3)√¤ -3¤ =3'2(cm) BCM=;2!;_6_3'2=9'2(cm¤ ) CE”="6¤√ +6¤ =6'2(cm)

CH”=3'2(cm) ACH

“AH="9√¤ -(√3√'2)¤ =3'7(cm) ( )=;3!;_6¤ _3'7

=36'7(cm‹ )

"1√0¤ -≈8≈¤ ='3å6=6

;3!;_(p_6¤ )_8=96p r

2pr=2p_6_ r=2(cm)

( )="6¤√ -2¤ =4'2(cm) AHO AH”="3¤√ -2¤ ='5(cm)

'5 cm

( )=p_('5)¤ =5p(cm¤ )

( )=DE”="3√¤ +9¤ ='9å0=3'1å0(cm)

( )=“AB

="(√8p)¤ √+(6çp)Ω¤

="1ç00ç≈p¤ =10p 120360

'32

'32 AD”=x

AB”=BC”="√x¤ +√3¤ +≈4Ω¤ ="√x¤ +ç25, AC”=x+x=2x , ABC=90 AB”¤ +BC”¤ =AC”¤

(x¤ +25)+(x¤ +25)=(2x)¤ , x¤ =25 x=5( x>0) AC”=2x=10 D’M”=D’N”="1√6¤ +ç8¤ =8'5(cm) M”N”="8√¤ +8¤ =8'2(cm)

DMN

( )="(√8'5√)¤ -√(4'ç2)Ω¤ =12'2(cm) DMN=;2!;_8'2_12'2=96(cm¤ )

AQ”

AQ”= _4=2'3(cm) QH”="(√2'3)√¤ -1¤ ='1å1(cm)

ABPQ=;2!;_(2+4)_'1å1

=3'1å1(cm¤ )

( )

=( )

=2p_2=4p(cm)

2p_6_ =4p ( )=120

BAC 2 '3=6 AC” AC”=3'3(cm) BA'C 2 '3=6 A’'C” A’'C”=3'3(cm)

( )=A’A'”=AC”+A’'C”=6'3(cm) 360

'32

( B-AFC )=( C-ABF

)

C-ABF ABF

BC”

C-ABF ;3!;_;2!;_2¤ _2=;3$;

( B-AFC )=;3$;

AFC 2'2

AFC _(2'2)¤ =2'3

B-AFC ;3$;

;3$;=;3!;_2'3_B’I’ B’I’= 2'33 '34

M

B C

6 cm 3'3 cm 3'3 cm

H

6 cm 9 cm

6 cm A

C

D

E

B

A 8p

6p A

D C G H

B F E

3 cm

3 cm 3 cm 3 cm

Q

A H B

2 P

4 2 2'3

1 1

B

A 30 C 30 A'

60 60

6 cm 6 cm

(17)

AO”=CO”=10 cm OH”=18-10=8(cm) OHC HC”="1√0¤ √-8¤ ="3Ω6=6(cm) ( )=;3!;_(p_6¤ )_18=216p(cm‹ )

x

2p_4=2p_16_

x=90

( )=B’M”="1√6¤ +ç8¤ =8'5(cm) x

360

86 60

57 kg 59

16

2'2å1 km cm¤

3'4å1

3'3 cm 256'5p

3

=6 a+b+c+d=8

a, b c, d

=;4*;=2

12, 13, 14, 15, 15, 16, 17, 18, 18, 19

=15.5( )

4, 7, a 7 aæ7

11, 13, a 11 a…11

7…a…11 x

=84, 334+x=420 x=86( )

82 , 86 , 72 , 86 , 94 86

82+x+72+86+94 5

15+16 2 a+b+c+d

4

(4a-2)+(4b-2)+(4c-2)+(4d-2) 4

80 x=80( )

, 4 =80( )

5 a 5

=80-4, 320+a=380 a=60( )

B x kg

4+x+(-2)+1+(-6)=0 x=3(kg)

B 3+54=57(kg)

=5 x=3

( )= = º;;=2

3 8

(A )= ∆º;;= ( )

(B )= ∆;;= ( )

(C )= ™∆;;='2( )

A, C, B

=8 x+y=15

=4 x¤ +y¤ =107 xy=59

( )=

=;1%0);=5( )

( )

=

=Æ;1%¬0*;='5ß.8( )

(A )=(B )= ∞;;=7( )

(B )

= ='2( )

(A )

= =Æ;5@;= ( )

B A

'1å05 1¤ +0¤ +0¤ +(-1)¤ +0¤

5

0¤ +(-2)¤ +1¤ +2¤ +(-1)¤

5

(-3)¤ _2+(-1)¤ _4+1¤ _2+3¤ _1+5¤ _1 10

2_2+4_4+6_2+8_1+10_1 10

(-3)¤ +0¤ +(x-8)¤ +(y-8)¤ +4¤

5 5+8+x+y+12

5

2'6 3 '1å53

(-1)¤ +(-2)¤ +2¤ +0¤ +1¤

5 4+x+7+5+6

5

90+70+80+80+a 5

90+70+80+80 4

æ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠

æ≠ ≠ ≠ ≠ ≠ ≠ ≠

æ≠ ≠ ≠ ≠ ≠ ≠

B

A

16 cm 8 cmM

4 cm

(18)

ADC x¤ +6¤ =10¤ x=8( x>0) ABC 17¤ =x¤ +(y+6)¤ , 289=64+(y+6)¤

y¤ +12y-189=0, (y+21)(y-9)=0 y=9( y>0)

x+y=17

ABCD '6å4=8(cm)

, AB”=BC”=CD”=DA”=8(cm) AH”=8-3=5(cm)

AEH EH”="3√¤ +5¤ ='3å4(cm) EFGH=('3å4)¤ =34(cm¤ )

x-4<x<x+4 x+4

(x+4)¤ =(x-4)¤ +x¤ , x¤ -16x=0 x(x-16)=0 x=16( x>4)

ABC BC”="8√¤ +6¤ ='1∂00=10 AB”¤ =BD”_BC”

8¤ =BD”_10 BD”=6.4 4¤ +OC”¤ =6¤ +8¤ , OC”¤ =84

OC”=2'2å1(km)( OC”>0)

S¡+S™=S£ S¡+S™=;2!;_p_8¤ =32p(cm¤ )

BDC™ BDE CBD= EBD

AD” BC” FDB= CBD( )

FBD= FDB BFD BF”=DF”

AF”=x DF”=BF”=10-x

ABF (10-x)¤ =x¤ +6¤ , 20x=64 x= (cm)

ABF=;2!;_AB”_AF”=;2!;_6_ (cm¤ )

"6√¤ +6¤ ='7å2=6'2(cm)

3'2 cm 3 cm

( )=p_(3'2)¤ -p_3¤

=18p-9p=9p(cm¤ )

A BC” H

BH”=15-10=5

ABH AH”=DC”="1√3¤ -ç5¤ ='1∂44=12 DBC BD”="1√5¤ +√12¤ =3'4å1

a 6

6_{ _a¤}=18'3, a¤ =12 a=2'3(cm)( a>0)

'34

A BC” H

BH”=x CH”=6-x

AH”¤ =7¤ -x¤ =5¤ -(6-x)¤ x=5 AH”="7√¤ -5¤ ='2å4=2'6

ABC=;2!;_6_2'6=6'6 ABC AC”="6√¤ +6¤ ='7å2=6'2

ACE 6'2 CE”='3 2 CE”=4'6 DCE CD” 4'6=1 '2 CD”=4'3 y=-(x¤ -6x+9-9)-2=-(x-3)¤ +7 P(3, 7)

x=0 y=-2 Q(0, -2)

PQ”="(√0-3√)¤ +√(-√2√-√7≈)¤ ='9å0=3'1å0 EG”="1√2¤ +ç8¤ ='2∂08=4'1å3(cm)

EP”=;2!;EG”=;2!;_4'1å3=2'1å3(cm)

AEP AEP=90

AP”="1√0¤ +√(2'1çå3)Ω¤ ='1∂52=2'3å8(cm) BD”=BG”=DG”=9'2 cm

BGD 9'2 cm

BGD= _(9'2)¤ = (cm¤ )

C BGD x cm

D-BCG

;3!;_ _x=;3!;_{;2!;_9_9}_9 x=3'3(cm)

( ABC )= _12¤ =36'3(cm¤ )

r

2pr=2p_12_ r=8

h

h="1√2¤ -ç8¤ ='8å0=4'5

( )=;3!;_p_8¤ _4'5= p

( )

=AD”'”

="(√4+3√+5)√¤ +8¤

='2∂08

=4'1ß3(cm)

256'5 3 240

360 '34 81'3

2

81'3 '3 2

4

8 cm 4 cm 3 cm

A

D E F D'

5 cm A'

B C

h 12

r

(19)

sin A=;5#;, cos A=;5$;, tan A=;4#; '6 6

1

^-1

1

^-2

2

^-1

2

^-2 ^;3*;

3

^-1 ^;5&;

3

^-2

4

^-1 ⁴

4

^-2 ^'2

5

^-1

5

^-2 ³⁽¹⁺^'3)

6

^-1

6

^-2 ^1.2738

7

^-1

7

^-2 ^-1 ¹

8

^-1

1.3724 40

8

^-2 ^0.1344

'53

;4#;

5'3 101

3+'5 ;3*;

2

1

^-1 ^{tan A=} ⁼ ^™;;

1

^-2^AC”="6√¤ -4¤ ='2å0=2'5 cos A= =

2

^-1

BC”="7√¤ -5¤ ='2å4=2'6 tan A=

2

^-23 cos A-1=0 cos A=;3!;

BC”="3√¤ -1¤ =2'2

sin A_tan A= _2'2=;3*;

3

^-1^BC”="8√¤ +6¤ =10

ABHª CBA(AA )

BCA= BAH= x

sin x= =;5#;, cos x= =;5$;

sin x+cos x=;5#;+;5$;=;5&;

3

^-2 ^{ACDª DCH} ^{x= DAC}

AC”="1√6¤ +ç12Ω¤ ='∂40å0=20 sin x= =;2!0@;=;5#;

4

^-1 ⁽ ^)=2_ ^_'3+4_;2!;_;2!;=3+1=4

4

^-2cos (x+15 )=;2!; x+15 =60 x=45

sin x+cos x=sin 45 +cos 45 = +'2='2 '2 2

2 '32

CD”

AC”

BC”

BA”

BC”

2'2 3 2'6

5

'53 2'5

6 BC”

AC”

5

^-1 ^{sin 30 =};1”2;=;2!; x=6 cos 30 =;1’2;= y=6'3

xy=6_6'3=36'3

5

^-2 ^ABH ^{sin 60 =} ⁼ ^AH”=3^'3

cos 60 = =;2!; BH”=3

AHC tan 45 = =1 CH”=3'3 BC”=BH”+CH”=3+3'3=3(1+'3)

6

^-1 cos 40 =0.77 cos 50 =0.64 tan 40 =0.84 tan 50 =1.19

6

^-2 ^{sin 35 =} =0.5736 tan 35 = =0.7002 sin 35 +tan 35 =1.2738

7

^-1 ^{sin 90 =1} ^{, tan 90}

7

^-2 ⁽ )=(0-1)(1-0)=(-1)_1=-1 ( )=1¤ +1¤ -1¤ =1+1-1=1

8

^-1 sin 39 =0.6293, cos 42 =0.7431 ( )=0.6293+0.7431=1.3724

tan 40 =0.8391 x=40

8

^-2 tan 42 =0.9004, cos 40 =0.7660 tan 42 -cos 40 =0.9004-0.7660

=0.1344

CD”

OD”

AB”

OA”

3'3 CH”

BH”6

'32 AH”6 '32

A C

B 7

5 2'6

A B

C

1

3 2'2

AB”="5√¤ +√1≈2¤ ='1∂69=13 sin A=;1!3@;, cos A=;1∞3;

sin A-cos A=;1!3@;-1∞3;=;1¶3;

BC”="5√¤ -4¤ ='9=3 tan A=;4#;

ABCª ACDª CBD x= B, y= A

A

B 90 -A C 5

3 4

(20)

BC”="1√7¤ -ç8¤ ='2∂25=15(cm) cos x=cos B=;1!7%;, cos y=cos A=;1•7;

cos x-cos y=;1!7%;-;1•7;=;1¶7;

3x+4y-12=0

x 4, y 3

AB” ="3√¤ +4¤ =5 sin a=;5#;

EG”="3√¤ +3¤ =3'2, “CE="(√3'2)¤√ +3¤ ='2å7=3'3

cos x= = = =

;2!;= _ = =

{;2!;}¤ +{;2!;}¤ =;2!;+1 ;2!;+ = +1

ABC tan 60 = AC”=10 tan 60 =10'3

ACD sin 30 = AD”=10'3 sin 30 =5'3

ABD tan 30 = = BD”=6'3

ACD tan 60 = ='3 CD”=2'3 BC”=BD”-CD”=6'3-2'3=4'3

x sin x .

;2!; '3 1 0

sin 50 =0.7660 x=50 tan 51 =1.2349 y=51

x+y=50 +51 =101

cos 51 = =0.6293 AB”=6.293

QC”=PC”=AP”=6 cm CR”=AB”=4 cm

CQR

QR”="6√¤ -4¤ =2'5(cm) H”A”=QB”=QR”=2'5 cm PH”=(6-2'5)=2(3-'5) cm

HQP

tan x= = =

= 2(3+'5) = 3+'52 (3-'5)(3+'5)

2 3-'5 4

2(3-'5) HQ”

PH”

AB”10 '32

6 CD”

'33 6 BD”

AD”

10'3 AC”10

1+'3 '3 2

2 '33 '31 '33 '33

'32

'63 '2'3 3'2 3'3 EG”

CE”

sin x= =;3!; AC”=9 ABCª EDC(AA ) 9 3=3 CD” CD”=1

CDE DE”="3√¤ -1¤ =2'2 tan y= =

ABC

tan 60 = = ='3 BC”=4'3(cm) DBC

sin 45 = = = BD”=4'6(cm)

45 <x<90 , sin x>cos x sin x+cos x>0, cos x-sin x<0 ( )=sin x+cos x+(cos x-sin x)

=2 cos x

2 cos x=;3@; cos x=;3!;

sin x_tan x=2'2_2'2=;3*;

3 '22 4'3 BD”

BC”

BD”

BC”4 BC”

AB”

'25 2'2

10 3 AC”

x 4, y -5 6('3-1) cm

9'4å1 8'2 41

3

æ≠ ≠ ≠ ≠

A

B 4 O

3

a y

x 5

A

B Q

R C P D

6 cmH

4 cm x

A

B x C

1 2'2 3

A BCD H

A’M”=D’M”= _4=2'3 M”H”=;3!;D’M”=;3!;_2'3=

AH”= (2'3)¤ -{ }¤ = , sin x= = ÷2'3=

tan x= = ÷ =2'2

sin x+tan x= +2'2=

-5x+4y+20=0 y=;4%;x-5

y=;4%;x-5 y=0 x=4

A(4, 0)

y=;4%;x-5 x=0 y=-5 B(0, -5)

x 4, y -5 .

8'2 2'2 3

3 2'3 4'6 3 AH” 3 MÚH”

2'2 4'6 3

AH” 3 A’M”

4'6 2'3 3

3

2'3 3 '32

(21)

AOB AB”='4ƒ¤ +5¤ ='4å1

sina= =

cosa= =

sina+cos a= + =

BEF=90 -45 =45 , CEF=90 -30 =60 EF”=x cm

BEF tan 45 = BF”=x tan 45 =x(cm)

CEF tan 60 = CF”=x tan 60 ='3x(cm)

BF”+CF”=BC” x+'3x=12, (1+'3)x=12

x= = =6('3-1)

EF” 6('3-1) cm 12(1-'3) (1+'3)(1-'3) 12

1+'3

CF”x BF”x

9'4å1 4'4å1 41 5'4å1 41

41 4'4å1 4 41

'4å1 5'4å1 5 41

'4å1

AC”=5 cm, BC”=5'3 cm 12'3 27'3

1

^-1

1

^-2 ⁶^{'3 m}

2

^-1

2

^-2

3

^-1

50(3-'3) m

3

^-2 ^9(3-^'3)

4

^-1

4

^-2

5(1+'3) cm

5

^-1 ²⁰^{'3 cm¤}

5

^-2 ¹²⁰

6

^-1

6

^-2

1

^-1 ^{sin 50 =} BC”=AB” sin 50

1

^-2AB”=6 tan 30 =6_ =2'3(m)

AC”= =6÷ =4'3(m)

( )=AB”+AC”=2'3+4'3=6'3(m)

2

^-1 ^AH”=3'2 sin 45 =3'2_ =3

ABH BAH=45 BH”=AH”=3

'22 '32 cos 306

'33 BC”

AB”

CH”=BC”-BH”=5-3=2 AHC AC”="√3¤ √+2¤ ='1å3

2

^-2 ^ABC A=180 -(30 +105 )=45

C AB” H

BCH CH”=12sin 30 =6(cm)

AHC AC”= =6'2(cm)

3

^-1 ^AH”=h

BH”=h tan (90 -45 )=h tan 45 =h CH”=h tan (90 -60 )=h tan 30 = h BC”=BH”+CH”

h+ h=100, { }h=100 h= =50(3-'3)(m)

3

^-2 ^BH”=h AH”=h tan 30 = h CH”=h tan 45 =h

AC”=AH”+CH” h+h=18

h= =9(3-'3)

4

^-1 ^AH”=h

BH”=h tan 60 ='3h CH”=h tan 30 = h BC”=BH”-CH”

12='3h- h, h=12

h=12_ =6'3

4

^-2 ^AH”=h

BH”=h tan 60 ='3 h, CH”=h tan 45 =h BC”=BH”-CH”=('3 -1)h=10

h= =5(1+'3 )(cm)

5

^-1 ^ABC=;2!;_8_10_sin 60 =20'3(cm¤ )

5

^-2 ^ABC=;2!;_8_9_sin (180 -C)=18'3 sin (180 -C)= 180 -C=60

C=120

6

^-1

ABCD=6_6_sin(180 -135 )=18'2

6

^-2 ^ABCD=;2!;_5_4_sin (180 -135 )=5'2 '32

10 '3-1

3 2'3

2'3 '3 3

3 '33 3+54'3

'33

'33 300

3+'3

3+'3 '3 3

3

'33 sin 456

A O

B

a 4 '4å1 5

(22)

4(1+'3) cm 75('3-1) m

12p-9'3

1.76 m ;;5#;

100'3 m 50 m 50'1å3 m (2+'3) cm 2-'3 40'3 cm¤

3 BC”=100 tan 31 =60(m)

( )=1.5+60=61.5(m) DH”=AH”=BD”=50(m)

CH”=AH ” tan 30 =50_ = (m)

“CD=“CH+“DH= +50= (m)

A BC” H

AH”=8 sin 60 =4'3(cm), BH”=8 cos 60 =4(cm) CH”= =4'3(cm)

“BC=BH”+CH”=4+4'3=4(1+'3)(cm) A

H ,

“AH=h “BH=h

AHC ACH=30

CAH=60 “CH=h tan 60 ='3h

“BC=“BH+“CH=h+'3h, ('3+1)h=150 h= =75('3-1)(m)

ABC CBH=60 ACB=30

“AB=“BC=50'3(m)

“CH=50'3 sin 60 =50'3_ =75(m) cos B=;3!;

A’'C'”="3√¤ -1¤ ='8=2'2 sin B=

ABC=;2!;_8_9_sin B

=;2!;_8_9_ =24'2(cm¤ )

BD”

ABCD= ABD+ BCD

=;2!;_5_5_sin(180 -120 )+;2!;_5'3_5'3_sin60

= +75'3=25'3(cm¤ ) 25'3 4

4

2'2 3 2'2

3

'32 '3+1150

4'3 tan 45

50('3+3) 50'3 3

3

50'3 '3 3

3

( )=8_{;2!;_4_4_sin 45 }=32'2 AMC=;4!; ABCD=;4!;_(4_3_sin 60 )=

“AC=“BD ABCD=;2!;_AC”_BD”_sin(180 -120 )

=;2!;_BD”¤ _sin60 =16'3 BD”¤ =64 BD”=8(cm)( BD”>0) OC”

( )

=( AOC )- AOC

=p_6¤ _ -;2!;_6_6_sin(180 -120 )

=12p-9'3 OH”=OC”_cos 45

=3.2_ =2.24(m) h=4-2.24=1.76(m)

B AC”

H

“BH=100sin 45 =50'2(m)

“AB= =

= (m)

AH”=CH” cos 60 = = CH”=200 cos 60 =200_;2!;=100(m) D’M”=D’N”="2√¤ +1¤ ='5

ABCD= AMD+ DMN+ DNC+ MBN

2_2=;2!;_2_1+;2!;_'5_'5_sinx+;2!;_1_2+;2!;_1_1 4=;2%;+;2%; sin x, ;2%; sin x=;2#; sin x=;5#;

CH”

200 AH”

200 100'6

3

50'2 cos 30 cos 30BH”

'22 120 360

3'3 2

ABH AH”=200 sin 60 =100'3(m) ABH BH”=200 cos 60 =100(m)

CH”=BC”-BH”=150-100=50(m)

150 m 45

45 30

B H

h

C A

A'

C' B

3

1

30

45 45

A

B C

H

100 m

O D

B A C

H 45 45 3.2 3.2

0.8 h

4

(23)

ACH

AC”="(√100√'3)¤√ +√50¤ ='3∂25∂00 =50'1å3(m)

CDB BD”= ='3(cm)

CD”= =2(cm)

AB”=AD”+BD”=CD”+BD”=2+'3(cm) CDB=30

CAD= ACD=;2!;_30 =15 CAB

tan 15 = = =2-'3 DCH

CD”= = (cm)

BH'C

BC”= = (cm)

ABCD

ABCD= _ _sin (180 -120 )

=40'3(cm¤ ) 3

10'3 8'3 3

3 8'3 4 3

sin 60

10'3 5 3

sin 60

1 2+'3 BC”

AB”

sin 301

1 tan 30

8 cm 8 cm 6 cm 18 cm

1

^-1

1

^-2 ^{20 cm}

2

^-1 ^{'1å1 cm}

2

^-2

3

^-1

5 cm

3

^-2 ⁸^'3

4

^-1 ^{3 cm¤}

4

^-2 ^{4 cm}

5

^-1

5

^-2 ³^{'3 cm}

6

^-1

6

^-2

7

^-1 ¹³

7

^-2

1

^-1

CE”+2AB”

1

^-2^OD” ^{AD” OC”}

OAD= BOC=30

AO”=OD” OAD= ODA=30

AOD=180 -2_30 =120

120 30 =µAD 5 µAD=20(cm)

2

^-1 ^AH”=;2!;AB”=5(cm)

OAH OH”="6√¤ -5¤ ='1å1(cm)

2

^-2 ^M”B”=;2!;AB”=6(cm) OB”=x O’M”=x-4

OBM x¤ =6¤ +(x-4)¤ , 8x=52 x=6.5(cm)

3

^-1 ^{O’M”=ON”} AB”=CD”=6(cm) CN”=;2!;CD”=3(cm)

OCN OC”="3√¤ +4¤ =5(cm)

3

^-2 ^OA”

OA”=4 A’M”="4√¤ -2¤ =2'3 AB”=2A’M”=4'3

O’M”=ON” CD”=AB”=4'3 AB”+CD”=4'3+4'3=8'3

4

^-1 PAO= PBO=90

AOB=360 -(90 +90 +60 )=120

p_3¤ _ =3p(cm¤ )

4

^-2 ^PT ^O ^{OT” PT”}

PA”=x

6¤ +8¤ =(6+x)¤ , x¤ +12x-64=0

(x+16)(x-4)=0 x=4(cm)( x>0)

5

^-1 ^PBO=90

PBO PB”="1√5¤ -≈9¤Ω =12(cm) PA”=PB”=12(cm)

PA”+PB”=24(cm)

5

^-2 ^OP” ^{AOP™ BOP}

APO= BPO=30 , AOP= BOP=60 AOP AP” OA”='3 1 9 OA”='3 1 OA”= =3'3(cm)

O 3'3 cm

6

^-1 ^{BP”=BQ”=x}

AP”=AR”=8-x, CQ”=CR”=10-x AC”=AR”+CR” 6=(8-x)+(10-x) 2x=12 x=6(cm)

6

^-2 ^O ^{r cm}

AD”=AF”=(8-r) cm CE”=CF”=(15-r) cm AC”="8√¤ +1≈5Ω¤ =17(cm) AC”=AF”+CF”=AD”+CE”

17=(8-r)+(15-r) r=3(cm) '39

120360

4 cm

5 cm

C H B

H' A D

60

A C

O

M B

x-4 x 46

(24)

7

^-1AB”+CD”=AD”+BC”

AB”+6=12+7 AB”=13

7

^-2 ^POSD, ^ORCS

DS”=SC”=2(cm)

AP”=x cm DP”=DS”=2(cm) 5+4=(x+2)+6 x=1(cm)

9 cm

4'5 2 cm

4.2 cm 20(2-'3) cm

DO”=DE” EOD= OED=25

ODE ODC=25 +25 =50

OC”=OD” OCD= ODC=50

OCE AOC=50 +25 =75 75 25 =µAC 3 µAC=9(cm) OB”=x

OH”=x-6, AH”=BH”=8

OHB x¤ =(x-6)¤ +8¤

12x=100 x= ™3∞;;(cm) OP”=PQ”=6(cm)

OAP AP”="1√2¤ -≈6Ω¤ ='1∂08=6'3(cm) AB”=2AP”=2_6'3=12'3(cm)

O’M”=ON”=2 AB”=CD”

AOM A’M”="√(2'2ç)¤ ç-ç2¤ =2 CD”=AB”=2A’M”=4

ABC AB”=AC”

B= C=;2!;_(180 -40 )=70 AOB=180 -60 =120

OAB OA”=OB”

x=;2!;_(180 -120 )=30

PO” AB” H

PO”="1√0¤ +≈5Ω¤ ='1ß2å5=5'5 APO PO” AB”

;2!;_PA”_OA”=;2!;_PO”_AH”

;2!;_10_5=;2!;_5'5_AH” AH”= =2'5 AB”=2AH”=4'5

'510

D AC” H

CD”=12+5=17(cm), CH”=12-5=7(cm) CHD DH”="1√7¤ -≈7Ω¤ =4'1å5(cm)

AB”=DH”=4'1å5(cm)

BD”=BE”=7(cm), AD”=AF”=10-7=3(cm) CF”=CE”=8-3=5(cm)

BC”=BE”+CE”=BD”+CF”=7+5=12(cm)

O r

(r+10)¤ +(r+3)¤ =13¤ , r¤ +13r-30=0 (r+15)(r-2)=0 r=-15 r=2

r=2(cm)( r>0)

AB”+CD”=AD”+BC”=3+7=10(cm) AB” CD”=3 2 2AB”=3CD

CD”=;3@;AB”

AB”+;3@;AB”=10, ;3%;;AB”=10 AB”=10_;5#;=6(cm)

CF”=C’I’=2(cm) EH”=E’I’=x

BE”=7-(2+x)=5-x, AE”=5+x ABE 4¤ +(5-x)¤ =(5+x)¤

x=0.8(cm)

BE”=5-0.8=4.2(cm)

AB” M , OA”, O’M”

O’M” AB”, A’M”=B’M”

OA”,

O’M” , 16p cm¤

OA”¤p-O’M”¤ p=16p, (OA”¤ -O’M”¤ )p=16p OA”¤ -O’M”¤ =16

OAM

A’M”="√OA”¤ √-O’çM”¤ ='1å6=4(cm) AB”=2A’M”=8(cm)

OD”=OE”=OF” AB”=BC”=CA”=8'3 cm

ABC .

ABC=3 OBC

_(8'3)¤=3_{;2!;_8'3_OE”} OE”=4(cm) '34

(25)

10 cm 9.6 cm 12 8 cm (8-x) cm 10 cm

AOO' O’O'”="6√¤ +8¤ =10(cm) AOO' BOO'(SSS )

O'AB , O’'M”

AB” O’'M” . AOO'

6_8=10_A’M”, A’M”=4.8(cm) AB”=2A’M”=9.6(cm) CE”=CF”=x

AB”=(10-x)+(14-x)=12 x=6

O PQ” R

PR”=PF”, QR”=QE”

( PQC )=PQ”+QC”+CP”

=2CE”=2_6=12 CD”=DP” DP”=8(cm)

EP”=EB”=x(cm) AE”=(8-x)(cm) AED

(8+x)¤ =(8-x)¤ +8¤ , 32x=64 x=2(cm) DE”=DP”+EP”=8+2=10(cm)

BOE OB”="√4¤ +√(√4'3≈)Ω¤ =8(cm) ( O )=p_8¤ =64p(cm¤ ) BE”=x BD”=x, AE”=7+x

CF”=CD”=5-x, AF”=6+(5-x)=11-x AE”=AF” 7+x=11-x, 2x=4

x=2(cm)

AE”=AB”+BE”=7+2=9(cm) O'

x

(10-x)¤ +(20-x)¤

=(10+x)¤

x¤ -80x+400=0 x=40—20'3

x=20(2-'3)(cm)( x<10)

120 55 30 68

1

^-1 ⁵⁰ ¹⁰⁰

1

^-2

2

^-1

2

^-2

3

^-1

3

^-2 ²⁸

4

^-1

4

^-2 ⁶⁶

5

^-1 ⁶⁰

5

^-2

50

6

^-1

6

^-2 ¹⁸⁰

7

^-1 ⁶²

7

^-2

1

^-1 ^x=;2!; AOB=;2!;_100 =50

x=;2!;_(360 -160 )=100

1

^-2 AOB=2_65 =130

A, B PAO= PBO=90

APB=360 -(90 +90 +130 )=50

2

^-1 ^{D= C=40}

PBD APB= D+ x

70 =40 + x x=30

2

^-2 ^BR”

ARB= APB=46 , BRC= BQC=20 x= ARB+ BRC=46 +20 =66

3

^-1 ^{µ BC}

BDC= BAC=47

AC” ABC=90

x=180 -(90 +47 )=43

3

^-2 ^{DAC= x}

APD x=180 -(90 +62 )=28

4

^-1 ^{µAB=µ BC} ^{x= BAC=40}

4

^-2 BAC= BDC=32

µAB=µ BC ADB= BDC=32 ABD

x=180 -(32 +50 +32 )=66

5

^-1 CED= CAD=20

BAC CAD=6 3=2 1 BAC=2 CAD=2_20 =40

BAD= BAC+ CAD=40 +20 =60

5

^-2 µAB µCD= ACB CAD

2 6=25 CAD CAD=75

APC P+25 =75 P=50

6

^-1 µAB µ BC µ CA=3 4 5

C A B=3 4 5

C= _180 =45

6

^-2 ^x=;3!;_180 =60 y=;6!;_180 =30 z= x+ y=60 +30 =90

x+ y+ z=60 +30 +90 =180 3

3+4+5

A

O

O' B

D

C 30

10 x

20 10-x 10+x 20-x

(26)

20

60 200 45

6 cm

y=2_110 =220

x=;2!;_(360 -220 )=70 x+ y=70 +220 =290 BOC=2 BAC=2_68 =136

OBC OB”=OC”

x=;2!;_(180 -136 )=22 OA”

AOB=2 AEB=40 , AOC=2 ADC=152 BOC= AOC- AOB=152 -40 =112 OA”, OB”

AOB=360 -2 AQB=160

APBO PAO= PBO=90

x=360 -(90 +90 +160 )=20 µCD

CAD= CBD=45

BPC x+45 =80 x=35

BDC= BAC=58 , ACB= ADB=40 BCD DBC=180 -(58 +22 +40 )=60

BD” ADB=90

µ BC

BAC= BDC=34

ADC= ADB- BDC=90 -34 =56 µAC=µ BD DCB= ABC=24

x= PCB+ PBC=24 +24 =48

4 2= x 25 x=50

y=2 APB+2 BQC

=2_50 +2_25 =100 +50 =150 x+ y=50 +150 =200

7

^-1 ^ABC ACB=180 -(43 +75 )=62 A, B, C, D

x= ACB=62

7

^-2 ABD=180 -(85 +60 )=35 x= ABD=35

79 30 114 40 80

18p cm

BC” ABC=180 _;6!;=30 BCD=180 _;1¡2;=15

APC= ABC+ BCD=30 +15 =45 BAC= BDC=35

ACB=90 -35 =55 ADB= ACB

ADB+ ACB ADB+ ACB BDC=100 -65 =35 BAC+ BDC BDC= BAC= x

AQC ACD=30 + x

, PCD

x+(30 + x)=70 x=20

AE” AEB=90

DAE=;2!; DOE=;2!;_40 =20

ACE ACE=180 -(90 +20 )=70 APB=;2!;_220 =110

µ PA µ PB=2 3 PBA PAB=2 3 , PBA=;3@; PAB

PAB 110 + PAB+;3@; PAB=180

;3%; PAB=70 PAB=42

µAC ADC= x

µ BD DAB= y

APD x+ y= BPD=60

µAC µ BD 120

(µAC+µ BD )=2p_9_120 =6p(cm) 360

BAP= BQP=32 ( µ BP )

APB=90 APC=90 -43 =47

APC x= BAP+ APC=32 +47 =79 µDE=;3!;µAB DOE=;3!;_180 =60

x=;2!;_60 =30